Understanding and Pushing the Limits of the Elo Rating Algorithm

This work is concerned with the rating of players/teams in face-to-face games with three possible outcomes: loss, win, and draw. This is one of the fundamental problems in sport analytics, where the very simple and popular, non-trivial algorithm was proposed by Arpad Elo in late fifties to rate chess players. In this work we explain the mathematical model underlying the Elo algorithm and, in particular, we explain what is the implicit but not yet spelled out, assumption about the model of draws. We further extend the model to provide flexibility and remove the unrealistic implicit assumptions of the Elo algorithm. This yields the new rating algorithm, we call $\kappa$-Elo, which is equally simple as the Elo algorithm but provides a possibility to adjust to the frequency of draws. The discussion of the importance of the appropriate choice of the parameters is carried out and illustrated using results from English Premier League football seasons.